9000031008
Level:
B
Assuming \(x\in \mathbb{R}\),
solve the following equation.
\[
4x^{3} - 3x^{2} - x = 0
\]
\( \left \{-\frac{1}
{4};0;1\right \}\)
\(\{0;1;4\}\)
\( \{1;4\}\)
\( \{0\}\)
Higher Degree Equations and Inequalities
9000028307
Level:
B
Solve the following equation.
\[
x^{3} + 6x^{2} - 8x = 0
\]
\(0\),
\(- 3 -\sqrt{17}\),
\(- 3 + \sqrt{17}\)
\(0\),
\(3 -\sqrt{17}\),
\(3 + \sqrt{17}\)
\(0\),
\(- 3\),
\(\sqrt{
17}\)
\(0\),
\(3\),
\(-\sqrt{17}\)
9000031010
Level:
B
Identify a true statement on the following equation.
\[
x^{5} - x^{3} - 6x = 0
\]
The equation has three solutions in \(\mathbb{R}\).
The equation does not have solution in
\(\mathbb{R}\).
The equation has five solutions in \(\mathbb{R}\).
The equation has one solution in \(\mathbb{R}\).
9000028308
Level:
B
Solve the following equation.
\[
x^{4} - 20x^{2} + 99 = 0
\]
\(-\sqrt{11}\),
\(- 3\),
\(3\),
\(\sqrt{
11}\)
\(0\),
\(- 3 -\sqrt{17}\),
\(- 3 + \sqrt{17}\)
\(0\),
\(3 -\sqrt{17}\),
\(3 + \sqrt{17}\)
\(-\sqrt{17}\),
\(- 3\),
\(3\),
\(\sqrt{
17}\)
9000031002
Level:
B
One of the solutions of the following equation is
\(x = 2\). Find
the set of all solutions.
\[
x^{3} + 2x^{2} - 5x - 6 = 0
\]
\(\{ - 3;-1;2\}\)
\( \{ - 3;-1\}\)
\( \{ - 3;0;2\}\)
\(\{ - 1;2;3\}\)
9000028309
Level:
C
Find the sum of all real solutions of the following equation.
\[
x^{4} + x^{3} + x^{2} + x = 0
\]
\(- 1\)
\(0\)
\(5\)
\(6\)
9000029301
Level:
B
Find the solution set of the following inequality.
\[
\left (x - 1\right )\left (x - 2\right )\left (x - 3\right )\geq 0
\]
\(\left [ 1;2\right ] \cup \left [ 3;\infty \right )\)
\(\left (-\infty ;\infty \right )\)
\(\left (-\infty ;1\right )\cup \left (2;3\right )\)
\(\emptyset \)
\(\{0\}\)
9000028302
Level:
B
The following equation has a solution
\(x = 1\). Find
the sum of the remaining real solutions.
\[
x^{3} + 2x^{2} - x - 2 = 0
\]
\(- 3\)
\(- 1\)
\(0\)
\(2\)
9000028303
Level:
B
The following equation has a solution
\(x = -2\). Find
the sum of the remaining real solutions.
\[
x^{3} + 3x^{2} - 18x - 40 = 0
\]
\(- 1\)
\(1\)
\(0\)
\(4\)
9000028304
Level:
C
The following equation has solutions
\(x_1= 1\) and
\(x_2 = 3\). Find
the sum of the remaining real solutions.
\[
x^{4} - 12x^{3} + 47x^{2} - 72x + 36 = 0
\]
\(8\)
\(- 1\)
\(3\)
\(5\)